Related papers: Separation of variables in multi-Hamiltonian syste…
The symmetric top is a special case of the general top, and canonical Poisson structure on $T^*SE(3)$ is the common method of its description. This structure is invariant under the right action of $SO(3)$, but the Hamiltonian of the…
A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of Hamiltonian vector…
We perform variable separation in the quasi-potential systems of equations of the form $\ddot{q}=-A^{-1}\nabla k=-\tilde{A}^{-1}\nabla\tilde{k}${}, where $A$ and $\tilde{A}$ are Killing tensors, by embedding these systems into a…
In our previous papers [11,13] we showed that the Hamilton-Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how…
An equation is obtained to find the Lagrangian for a one-dimensional autonomous system. The continuity of the first derivative of its constant of motion is assumed. This equation is solved for a generic nonconservative autonomous system…
We review the theory of orthogonal separation of variables on pseudo-Riemannian manifolds of constant non-zero curvature via concircular tensors and warped products. We then apply this theory simultaneously to both the three-dimensional…
We review a perturbative approach to deal with Lagrangians with higher or infinite order time derivatives. It enables us to construct a consistent Poisson structure and Hamiltonian with only first time derivatives order by order in…
Motivated by the problem of longitudinal data assimilation, e.g., in the registration of a sequence of images, we develop the higher-order framework for Lagrangian and Hamiltonian reduction by symmetry in geometric mechanics. In particular,…
Topological constraints on a dynamical system often manifest themselves as breaking of the Hamiltonian structure; well-known examples are non-holonomic constraints on Lagrangian mechanics. The statistical mechanics under such topological…
We deal with Lagrangians which are not the standard scalar ones. We present a short review of tensor Lagrangians, which generate massless free fields and the Dirac field, as well as vector and pseudovector Lagrangians for the electric and…
For an integrable Hamiltonian system we construct a representation of the phase space symmetry algebra over the space of functions on a Lagrangian manifold. The representation is a result of the canonical quantization of the integrable…
We introduce natural differential geometric structures underlying the Poisson-Vlasov equations in momentum variables. We decompose the space of all vector fields over particle phase space into a semi-direct product algebra of Hamiltonian…
We consider nonholonomic systems which symmetry groups consist of two subgroups one of which represents rotations about the axis of symmetry. After nonholonomic reduction by another subgroup the corresponding vector fields on partially…
An outline of the basic Riemannian structures underlying the separation of variables in the Hamilton-Jacobi equation of natural Hamiltonian systems.
All equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces, provided that Hamilton-Jacobi equation and Klein-Gordon-Fock equation for a charged test particle can be integrated by the method of complete…
In this paper we explore general conditions which guarantee that the geodesic flow on a 2-dimensional manifold with indefinite signature is locally separable. This is equivalent to showing that a 2-dimensional natural Hamiltonian system on…
We discuss several new bi-Hamiltonian integrable systems on the plane with integrals of motion of third, fourth and sixth order in momenta. The corresponding variables of separation, separated relations, compatible Poisson brackets and…
Certain dissipative physical systems closely resemble Hamiltonian systems in $\mathbb{R}^{2n}$, but with the canonical equation for one of the variables in each conjugate pair rescaled by a real parameter. To generalise these dynamical…
We study concircular tensors in spaces of constant curvature and then apply the results obtained to the problem of the orthogonal separation of the Hamilton-Jacobi equation on these spaces. Any coordinates which separate the geodesic…
Starting from a bundle E over R, the dual of the first jet bundle, which is a co-dimension 1 sub-bundle of the cotangent bundle of E, is the appropriate manifold for the geometric description of time-dependent Hamiltonian systems. Based on…